Elastic wave-mode separation plays an important role in elastic reverse-time migration (RTM) and elastic full-waveform inversion. It helps to remove crosstalk artifacts and improve imaging quality. A class of efficient methods for elastic wave-mode separation in isotropic elastic media is the Helmholtz decomposition technique. Although this kind of approach produces pure-mode vector wavefields with correct amplitudes, phases, and physical units, their computational costs are still high especially for 3D large-scale problems. By making use of the relationships among the divergence, curl, gradient, and exterior derivative operations, we develop an improved elastic wave-mode separation based on the Helmholtz decomposition. We also need to solve a Poisson equation, but the Laplace operator operates on a scalar function rather than a vector function. Thus, for multidimensional (2D or 3D) problems, the Poisson equation only needs to be solved once for the vector P- and S-wavefields. This allows us to reduce the computational cost of the conventional Helmholtz decomposition method by a factor of 2 for solving 2D problems. For a 3D problem, the computational cost can be reduced by a factor of 3. To further reduce the computational cost, by introducing a smooth extension technique, we transform the problem into the wavenumber domain via Fourier transform and propose a fast solver for the Poisson equation. The resulting wavefield separation method not only produces P-wave and S-wave with the same phases and amplitudes as the input coupled wavefields but also significantly reduces the computational cost. Numerical tests show the efficiency of the proposed method and confirm the theoretical results.
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